OEIS A000035: Parity sequence

Join us as we dive into the parity sequence a(n) = n mod 2: 0, 1, 0, 1, ... We explore how this simple alternation hides surprising depth—from appearing in the decimal expansion 0.010101… of 1/99 and the binary expansion of 1/3, to a family of generating functions (ordinary, exponential, Dirichlet) that encode the entire infinite pattern. We connect the dots to concepts like the least-significant bit, Kronecker symbols, and the Riemann zeta function, and we discuss practical uses in error detection (parity bits), cryptography, and algorithm design. Drawing on OEIS entry A000035 and the document “Properties and Applications of the Parity Sequence” (A00003), this episode unveils how a tiny, everyday pattern threads through deep mathematics and real-world tech.

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